Generalized twoqubit whole and half HilbertSchmidt separability probabilities
Abstract
Compelling evidencethough yet no formal proofhas been adduced that the probability that a generic (standard) twoqubit state ($\rho$) is separable/disentangled is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$hypergeometric moment formula (Appendix A), we reach, {\it via} densityapproximation procedures, the conclusion that onehalf ($\frac{4}{33}$) of this probability arises when the determinantal inequality $\rho^{PT}>\rho$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $\rho>\rho^{PT}$. These probabilities are taken with respect to the flat, HilbertSchmidt measure on the fifteendimensional convex set of $4 \times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two"re[al]bit" and two"quater[nionic]bit"separability probabilities of $\frac{29}{64}$ and $\frac{26}{323}$, respectively. The new determinantal $4F3$hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states ($\rho=0$), and its consistency manifestedalso using densityapproximationwith a theorem of Szarek, Bengtsson and {Ż}yczkowski (arXiv:quantph/0509008). This theorem states that the HilbertSchmidt separability probabilities of generic minimally degenerate twoqubit states are (again) onehalf those of the corresponding generic nondegenerate states.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 April 2015
 DOI:
 10.1016/j.geomphys.2015.01.006
 arXiv:
 arXiv:1404.1860
 Bibcode:
 2015JGP....90...42S
 Keywords:

 Geometry of quantum states;
 Quantum information;
 2×2 quantum systems;
 Entanglement probability distribution moments;
 Probability distribution reconstruction;
 HilbertSchmidt measure;
 Quantum Physics;
 Mathematical Physics;
 Mathematics  Probability;
 81P45;
 33C90;
 62G07
 EPrint:
 25 pages, minor changes, to appear in the Journal of Geometry and Physics. (Paper expands substantially upon arXiv::1403.1825.)